Comparison of Variational, Differential Quadrature, and Approximate Closed-Form Solution Methods for Buckling of Highly Flexurally Anisotropic Laminates
Publication: Journal of Engineering Mechanics
Volume 139, Issue 8
Abstract
The buckling response of symmetric laminates that possess strong flexural-twist coupling is studied using different methodologies. Such plates are difficult to analyze because of localized gradients in the mode shape. Initially, the energy method (Rayleigh-Ritz) using Legendre polynomials is employed, and the difficulty of achieving reliable solutions for some extreme cases is discussed. To overcome the convergence problems, the concept of Lagrangian multiplier is introduced into the Rayleigh-Ritz formulation. The Lagrangian multiplier approach is able to provide the upper and lower bounds of critical buckling load results. In addition, mixed variational principles are used to gain a better understanding of the mechanics behind the strong flexural-twist anisotropy effect on buckling solutions. Specifically, the Hellinger-Reissner variational principle is used to study the effect of flexural-twist coupling on buckling and also to explore the potential for developing closed-form solutions for these problems. Finally, solutions using the differential quadrature method are obtained. Numerical results of buckling coefficients for highly anisotropic plates with different boundary conditions are studied using the proposed approaches and compared with finite-element results. The advantages of both the Lagrangian multiplier theory and the variational principle in evaluating buckling loads are discussed. In addition, a new simple closed-form solution is shown for the case of a flexurally anisotropic plate with three sides simply supported and one long edge free.
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References
Abaqus 6.10 [Computer software]. Vélizy-Villacoublay, France, Dassault Systèmes.
Ashton, J. (1969). “Analysis of anisotropic plates 2.” J. Compos. Mater., 3(3), 470–479.
Ashton, J., and Waddoups, M. (1969). “Analysis of anisotropic plates.” J. Compos. Mater., 3(1), 148–165.
Balabuch, L. (1937). “The stability of plywood plates.” Aeronaut. Eng. USSR, 11(9), 19–38.
Bellman, R. E., and Casti, J. (1971). “Differential quadrature and long-term integration.” J. Math. Anal. Appl., 34(2), 235–238.
Bhat, R. (1985). “Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method.” J. Sound Vibrat., 102(4), 493–499.
Budiansky, B., and Hu, P. C. (1946). “The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates.” NACA Rep. No. 848, National Advisory Committee for Aeronautics, Washington, DC.
Chamis, C. C. (1969). “Buckling of anisotropic composite plates.” J. Struct. Div., 95(10), 2119–2139.
Chien, W.-Z. (1984). “Generalized variational principles in elasticity.” Eng. Mech. Civ. Eng., 24, 93–153.
Chow, S., Liew, K., and Lam, K. (1992). “Transverse vibration of symmetrically laminated rectangular composite plates.” Compos. Struct., 20(4), 213–226.
Darvizeh, M., Darvizeh, A., Ansari, R., and Sharma, C. B. (2004). “Buckling analysis of generally laminated composite plates (generalized differential quadrature rules versus Rayleigh–Ritz method).” Compos. Struct., 63(1), 69–74.
Green, A., and Hearmon, R. (1945). “The buckling of flat rectangular plywood plates.” Philos. Mag., 36(261), 659–688.
Grenestedt, J. L. (1989). “Study on the effect of bending-twisting coupling on buckling strength.” Compos. Struct., 12(4), 271–290.
Herencia, J. E., Weaver, P. M., and Friswell, M. I. (2010). “Closed-form solutions for buckling of long anisotropic plates with various boundary conditions under axial compression.” J. Eng. Mech., 136(9), 1105–1114.
Liew, K., and Wang, C. (1995). “Elastic buckling of regular polygonal plates.” Thin-walled Struct., 21(2), 163–173.
Nemeth, M. P. (1986). “Importance of anisotropy on buckling of compression-loaded symmetric composite plates.” AIAA J., 24(11), 1831–1835.
Pandey, M. D., and Sherbourne, A. N. (1991). “Buckling of anisotropic composite plates under stress gradient.” J. Eng. Mech., 117(2), 260–275.
Plass, H. J., Gaines, J. H., and Newsom, C. D. (1962). “Application of Reissner’s variational principle to cantilever plate deflection and vibration problems.” J. Appl. Mech., 29(1), 127–135.
Reissner, E. (1950). “On a variational theorem in elasticity.” J. Math. Phys., 29(2), 90–95.
Sherbourne, A. N., and Pandey, M. D. (1991). “Differential quadrature method in the buckling analysis of beams and composite plates.” Comp. Struct., 40(4), 903–913.
Shu, C. (2000). Differential quadrature and its application in engineering, Springer, London.
Smith, S. T., Bradford, M. A., and Oehlers, D. J. (1999). “Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh–Ritz method.” Int. J. Numer. Methods Eng., 44(11), 1685–1707.
Tang, W.-Y., and Sridharan, S. (1990). “Buckling analysis of anisotropic plates using perturbation technique.” J. Eng. Mech., 116(10), 2206–2222.
Thielemann, W. (1950). “Contribution to the problem of buckling of orthotropic plates, with special reference to plywood.” NACA Technical Memorandum 1263, National Advisory Committee for Aeronautics, Washington, DC.
Washizu, K. (1975). Variational methods in elasticity and plasticity, 2nd Ed., Pergamon, London.
Weaver, P. M. (2006). “Approximate analysis for buckling of compression loaded long rectangular plates with flexural/twist anisotropy.” Proc. Royal Soc. London, Ser. A, 462(2065), 59–73.
Weaver, P. M., and Herencia, J. E. (2007). “Buckling of a flexurally anisotropic plate with one edge free.” 48th Proc., AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf., Vol. 8, 8534–8542.
Weaver, P. M., and Nemeth, M. P. (2007). “Bounds on flexural properties and buckling response for symmetrically laminated composite plates.” J. Eng. Mech., 133(11), 1178–1191.
Whitney, J. (1972). “Free vibration of anisotropic rectangular plates.” J. Acoust. Soc. Am., 52(1B), 448–449.
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Published In
Journal of Engineering Mechanics
Volume 139 • Issue 8 • August 2013
Pages: 1073 - 1083
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Oct 1, 2011
Accepted: Jul 25, 2012
Published online: Jul 28, 2012
Published in print: Aug 1, 2013
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